Editing List of mathematical functions

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In [[mathematics]], some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of [[special functions]] which developed out of [[statistics]] and [[mathematical physics]]. A modern, abstract point of view contrasts large [[function space]]s, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as [[symmetry]], or relationship to [[harmonic analysis]] and [[group representation]]s.

See also [[List of types of functions]]

==Elementary functions==
[[Elementary functions]] are functions built from basic operations (e.g. addition, exponentials, logarithms...)

===Algebraic functions===
[[Algebraic function]]s are functions that can be expressed as the solution of a polynomial equation with integer coefficients.
* [[Polynomial]]s: Can be generated solely by addition, multiplication, and raising to the power of a positive integer.
** [[Constant function]]: polynomial of degree zero, graph is a horizontal straight line
** [[Linear function]]: First degree polynomial, graph is a straight line.
** [[Quadratic function]]: Second degree polynomial, graph is a [[parabola]].
** [[Cubic function]]: Third degree polynomial.
** [[Quartic function]]: Fourth degree polynomial.
** [[Quintic function]]: Fifth degree polynomial.
* [[Rational function]]s: A ratio of two polynomials.
* [[nth root|''n''th root]]
** [[Square root]]: Yields a number whose square is the given one.
** [[Cube root]]: Yields a number whose cube is the given one.

===Elementary transcendental functions===
[[Transcendental function]]s are functions that are not algebraic.
* [[Exponential function]]: raises a fixed number to a variable power.
* [[Hyperbolic function]]s: formally similar to the [[trigonometric functions]].
** [[Inverse hyperbolic functions]]: [[Inverse function|inverses]] of the [[hyperbolic functions]], analogous to the [[Inverse trigonometric functions|inverse circular functions]].

* [[Logarithm]]s: the inverses of exponential functions; useful to solve equations involving exponentials.
** [[Natural logarithm]]
** [[Common logarithm]]
** [[Binary logarithm]]
* [[Exponentiation#Power functions|Power functions]]: raise a variable number to a fixed power; also known as [[Allometric function]]s; note: if the power is a rational number it is not strictly a transcendental function.
* [[Periodic function]]s
** [[Trigonometric function]]s: [[sine]], [[cosine]], [[tangent (trigonometry)|tangent]], [[cotangent]], [[secant (trigonometry)|secant]], [[cosecant]], [[exsecant]], [[excosecant]], [[versine]], [[coversine]], [[vercosine]], [[covercosine]], [[haversine]], [[hacoversine]], [[havercosine]], [[hacovercosine]], [[Inverse trigonometric functions]] etc.; used in [[geometry]] and to describe periodic phenomena. See also [[Gudermannian function]].

==Special functions==
{{main|Special functions}}
===Piecewise special functions===
{{columns-list|colwidth=20em|
* [[Indicator function]]: maps ''x'' to either 1 or 0, depending on whether or not ''x'' belongs to some subset.
* [[Step function]]: A finite [[linear combination]] of [[indicator function]]s of [[half-open interval]]s.
** [[Heaviside step function]]: 0 for negative arguments and 1 for positive arguments. The integral of the [[Dirac delta function]].
* [[Sawtooth wave]]
* [[Square wave (waveform)|Square wave]]
* [[Triangle wave]]
* [[Rectangular function]]
* [[Floor function]]: Largest integer less than or equal to a given number.
* [[Ceiling function]]: Smallest integer larger than or equal to a given number.
* [[Sign function]]: Returns only the sign of a number, as +1, &minus;1 or 0.
* [[Absolute value]]: distance to the origin (zero point)
}}

===Arithmetic functions===
{{main|Arithmetic function}}
* [[divisor function|Sigma function]]: [[Summation|Sums]] of [[Exponentiation|power]]s of [[divisor]]s of a given [[natural number]].
* [[Euler's totient function]]: Number of numbers [[coprime]] to (and not bigger than) a given one.
* [[Prime-counting function]]:  Number of [[prime number|prime]]s less than or equal to a given number.
* [[Partition function (number theory)|Partition function]]:  Order-independent count of ways to write a given positive integer as a sum of positive integers.
* [[Möbius function|Möbius μ function]]: Sum of the nth primitive roots of unity, it depends on the prime factorization of n.
* [[Prime omega function]]s
* [[Chebyshev function]]s
* [[Liouville function]], λ(''n'') = (–1)<sup>Ω(''n'')</sup>
* [[Von Mangoldt function]], Λ(''n'') = log&nbsp;''p'' if ''n'' is a positive power of the prime ''p''
* [[Carmichael function]]

===Antiderivatives of elementary functions===
* [[Logarithmic integral function]]: Integral of the reciprocal of the logarithm, important in the [[prime number theorem]].
* [[Exponential integral]]
* [[Trigonometric integral]]: Including Sine Integral and Cosine Integral
* [[Inverse tangent integral]]
* [[Error function]]: An integral important for [[normal distribution|normal random variables]].
** [[Fresnel integral]]: related to the error function; used in [[optics]].
** [[Dawson function]]: occurs in [[probability]].
** [[Faddeeva function]]

===Gamma and related functions===
* [[Gamma function]]: A generalization of the [[factorial]] function.
* [[Barnes G-function]]
* [[Beta function]]: Corresponding [[binomial coefficient]] analogue.
* [[Digamma function]], [[Polygamma function]]
* [[Incomplete beta function]]
* [[Incomplete gamma function]]
* [[K-function]]
* [[Multivariate gamma function]]: A generalization of the Gamma function useful in [[multivariate statistics]].
* [[Student's t-distribution]]
* [[Gamma function#Pi function|Pi function]] <math>\Pi(z) = z \Gamma(z) = (z)!</math>

===Elliptic and related functions===
{{columns-list|colwidth=20em|
* [[Elliptic integral]]s: Arising from the path length of [[ellipse]]s; important in many applications. Alternate notations include:
** [[Carlson symmetric form]]
** [[Legendre form]]
* [[Nome (mathematics)|Nome]]
* [[Quarter period]]
* [[Elliptic function]]s: The inverses of elliptic integrals; used to model double-periodic phenomena.
**[[Jacobi's elliptic functions]]
**[[Weierstrass's elliptic functions]] 
**[[Lemniscate elliptic functions]]
* [[Theta functions]]
* [[Neville theta functions]]
* [[Modular lambda function]]
* Closely related are the [[modular form]]s, which include
** [[J-invariant]]
** [[Dedekind eta function]]
}}

===Bessel and related functions===
{{columns-list|colwidth=20em|
* [[Airy function]]
* [[Bessel function]]s: Defined by a [[differential equation]]; useful in [[astronomy]], [[electromagnetism]], and [[mechanics]].
* [[Bessel–Clifford function]]
* [[Kelvin functions]]
* [[Legendre function]]: From the theory of [[spherical harmonics]].
* [[Scorer's function]]
* [[Sinc function]]
* [[Hermite polynomials]]
* [[Laguerre polynomials]]
* [[Chebyshev polynomials]]
* [[Synchrotron function]]
}}

===Riemann zeta and related functions===
{{columns-list|colwidth=20em|
* [[Riemann zeta function]]: A special case of [[Dirichlet series]].
* [[Riemann Xi function]]
* [[Dirichlet eta function]]: An allied function.
* [[Dirichlet beta function]]
* [[Dirichlet L-function]]
* [[Hurwitz zeta function]]
* [[Legendre chi function]]
* [[Lerch transcendent]]
* [[Polylogarithm]] and related functions:
** [[Incomplete polylogarithm]]
** [[Clausen function]]
** [[Complete Fermi–Dirac integral]], an alternate form of the polylogarithm.
** [[Dilogarithm]]
** [[Incomplete Fermi–Dirac integral]]
** [[Kummer's function]]
* [[Riesz function]]
}}

===Hypergeometric and related functions===
* [[Hypergeometric function]]s: Versatile family of [[power series]].
* [[Confluent hypergeometric function]]
* [[Associated Legendre functions]]
* [[Meijer G-function]]
* [[Fox H-function]]

===Iterated exponential and related functions===

* [[Hyper operator]]s
* [[Iterated logarithm]]
* [[Pentation]]
* [[Super-logarithm]]s
* [[Tetration]]

===Other standard special functions===

* [[Lambert W function]]: Inverse of ''f''(''w'') = ''w'' exp(''w'').
* [[Lamé function]]
* [[Mathieu function]]
* [[Mittag-Leffler function]]
* [[Painlevé transcendents]]
* [[Parabolic cylinder function]]
* [[Arithmetic–geometric mean]]

===Miscellaneous functions===

* [[Ackermann function]]: in the [[theory of computation]], a [[computable function]] that is not [[primitive recursive function|primitive recursive]].
* [[Dirac delta function]]: everywhere zero except for ''x'' = 0; total integral is 1. Not a function but a [[distribution (mathematics)|distribution]], but sometimes informally referred to as a function, particularly by physicists and engineers.
* [[Dirichlet function]]: is an [[indicator function]] that matches 1 to [[Rational number|rational numbers]] and 0 to [[Irrational number|irrationals]]. It is [[nowhere continuous]].
* [[Thomae's function]]: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function.
* [[Kronecker delta function]]: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
* [[Minkowski's question mark function]]: Derivatives vanish on the rationals.
* [[Weierstrass function]]: is an example of [[continuous function]] that is nowhere [[Differentiable function|differentiable]]

== See also ==
* [[List of types of functions]]
* [[Test functions for optimization]]
* [[List of mathematical abbreviations]]
* [[List of special functions and eponyms]]

== External links ==
* [https://archive.today/20130105073730/http://www.special-functions.com/ Special functions] : A programmable special functions calculator.
* [http://eqworld.ipmnet.ru/en/auxiliary/aux-specfunc.htm Special functions] at EqWorld: The World of Mathematical Equations.

[[Category:Calculus|Functions]]
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